Let
R be a ring,
S a strictly ordered monoid and ω:
S → End(
R) a monoid homomorphism. The skew generalized power series ring
R[[
S, ω]] is a common generalization of skew polynomial rings, skew power series rings, skew Laurent polynomial rings, skew group rings, and Mal'cev-Neumann Laurent series rings. In the case where
S is positively ordered we give sufficient and necessary conditions for the skew generalized power series ring
R[[
S, ω]] to have weak dimension less than or equal to one. In particular, for such an
S we show that the ring
R[[
S, ω]] is right duo of weak dimension at most one precisely when the lattice of right ideals of the ring
R[[
S, ω]] is distributive and ω(
s) is injective for every
s ∈
S.
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